a list compiled by Alex Kasman (College of Charleston)
|Note: This work of mathematical fiction is recommended by Alex for children and young adults.|
I finally read this book (Feb 2001)! It really was very nice, and none of the reviews I had read previously prepared me for the beautiful color illustrations throughout. My personal favorite among the boy's dreams is the one in which he learns about both the "Golden Mean" and the Euler characteristic. Though both of these things can be presented in a much more sophisticated way, it is wonderful how they can be delivered so casually, as if you were just saying "Hey, look, if you get out your calculator or draw a few figures on paper, you can notice some cool things!" I was a bit troubled by the fact that the author renames things, making it difficult for the reader of this book to continue their reading elsewhere. (For instance, prime numbers are `prima donnas', irrational numbers are `unreasonable' and Felix Klein is `translated' into Dr. Happy Little.) However, when I got to the end of the book I found that there is an index which also retranslates these terms back into their more familiar forms and was greatly relieved.
Oh, hey! This is the first time I've heard about the software myself. Apparently there is a computer game based on the book. I'm tempted to buy a copy right now, just based on the fact that the book was so great. However, I'd be disappointed if it turns out to be nothing more than the book, read aloud and reproduced page by page. In particular, I'm hoping that it is very interactive and contains features that let curious kids go beyond what they may have already seen in the book. Can anyone write in to tell me (and Sue, above) how it is before we put our money down on it?
Dear Mary Jo:
On page 74 where Robert divides 7 by 11 and 6 by 7, he gets two numbers which (obviously) are rational. He notes that there is a repeating pattern to their decimal expansion (a repeating 36 in the former case and a repeating 857142 in the latter). This is when the Number Devil first introduces the idea of irrational numbers by saying that there are others that are worse than these (rational) ones:
He goes on to suggest that Robert take the square root ("rutabaga") of 2 and notes that he does not see any repeats in the decimal expansion. Now, one could argue that this is not a valid proof that the square root of 2 is irrational, since it could possibly just have a repeating pattern that is not apparent in the display of Robert's calculator (just as it is possible that the apparent pattern of 7/11 breaks down somewhere outside of the display on his calculator). But, he does not say that 7/11 or 6/7 are unreasonable...he is trying to draw a contrast between those rational numbers and the irrational square root of 2.
In the case of the ratios of Fibonacci terms, Robert is looking at a list and notes that 89/55 has a repeating 18 in its decimal expansion, but describes 21/13 as "looking as unreasonable as they come". Of course, 21/13 is not irrational, and perhaps Robert should know that, too. However, he doesn't actually say it is unreasonable, only that the decimal expansion looks unreasonable. In other words, this is an example, such as the type I warned about in the previous paragraph, which might look irrational on a calculator since the repeating pattern is not apparent. The author further confuses things by pointing out that the limit as n goes to infinity of the ratio of the n and n+1 term in the sequence actually is an irrational (unreasonable) number. I will agree that this may muddle things to the point that someone who is naive about number theory would become confused. However, looking at it myself now, I cannot see that he ever says anything that is actually wrong.
At least, that's the way it looks to me. I hope that helps!
(Note that this book is reviewed in the AMS Notices.)
Maria, that is a great question. I think I see exactly what you mean. There are parts of mathematics that clearly have an application (like computing your income taxes or landing a space ship on the moon) and other parts of math that seem more like puzzles serving no other purpose than the enjoyment of the mathematicians playing with them.
There are in fact many applications for Pascal's Triangle, because the numbers in its rows are the answers to two very practical types of questions. Before I get to that, however, I need to set up a sort of strange terminology. Let's say that the top of the triangle ("1") is the 0th row and that the next row ("1 1") is the 1st row and the next row ("1 2 1") is thesecond row and so on...since that will make it easier for me to explain. And similarly, let's call the 1 at the start of each row the "zeroeth" number in that row and the number after it the first number. So, according to this strange notation, the first number in the second row is 2...right? Okay. Now, I can show you an application of the triangle.
Suppose I have four socks (sock 1, sock 2, sock 3 and sock 4) in a drawer. If I reach in and without looking pick two socks, how many possibilities are there for which two socks I get? As you can see by counting all of the possibilities, there are SIX possible pairs because I could get 1&2, 1&3, 1&4, 2&3, 2&4, or 3&4...there are six of them. This is connected to the fact that the number in the second spot in the fourth row of the triangle is six!
In general, we can say that the number of ways of picking k objects out of a set of n objects is given by the number in the kth spot of the nth row of the triangle. This becomes more impressive when the numbers are larger and you cannot easily count all of the possibilities.
Suppose I have invited 20 guests to a party but only have enough seating for 10. How many different possibilities are there for the list of 10 people who get to sit? It would take too long to try to list all possible choices of 10 names of the 20 guests. Instead, we should use the triangle. The number of possible sets of 10 selected out of a larger set of 20 would be the number in the 10th spot in the 20th row of Pascal's Triangle: 184756.
This procedure is frequently needed by people computing probabilities. For instance: the odds of winning the lottery. If your lottery asks you to pick 5 numbers from 1 to 60, then your odds of getting the same 5 numbers as the ones that pop up on TV are 1 in 5461512 (that is, less than one in five million) because that is the number in the 5th spot of the 60th row of the table. It is because of this that Pascal's triangle can be described as having a similar significance as the famous "bell curve" that you have probably heard about. (See this other website for a more careful explanation of this idea.)
Another type of question that is answered by Pascal's Triangle has to do with algebra. Most algebra students know that (x+y)2=x2+2xy+y2. Note that the coefficients in this expansion are the same as the numbers in the second row of the triangle! More generally, you can read off the coefficients of (x+y)n from the nth row of the table. The fourth row is "1 4 6 4 1" which means that (x+y)4=x^4 + 4 x3y+6x2y2+4xy3+y4.
The Fibonacci series is not quite as broadly useful, but it also has connections to the real world. In particular, biology seems to have "discovered" the usefulness of Fibonacci numbers long before we did, which explains why we can find them so often when counting petals or seeds on plants. Specifically, we can show that the Fibonacci numbers arise when we try to figure out how to optimally pack things around in a circle, as plants want to do with their own parts. Furthermore, the Fibonacci numbers are related to the aesthetics espoused by the ancient Greeks (the Golden Ratio) and the ratios of male to female bees in bee colonies! For more detailed explanations of these ideas, please see here and here.
That's my answer to your question. But, before you leave, let me offer one more bit of philosophy. Again, I do understand that some bits of mathematics appear to be much more useful than other bits. Some things that mathematicians study and play with may appear to be totally useless. I would first like to argue that there is a value in exploring the "mathematical landscape" even if no obvious application is apparent...just like we consider it to be valuable for geographic explorers to look around at places that no person has visited just to see if there is anything useful or interesting there. But, most amazingly, I'd like to add that it has happened quite often that mathematicians were studying something that appeared to be completely useless, and it later turned out to have been incredibly useful in ways that nobody had imagined. D'Alembert studied the mathematics of how violin strings vibrate, which may seem unimportant since people were making perfectly good violins without him...but it turns out to have been an important step in our later discovery of radio waves. Imaginary numbers are frequently mentioned as being "unrealistic", but we use them to describe the light waves that we send through fiber optic cables in the latest advances in communications. Mathematicians in the 19th century studied "non-Euclidean geometry", which I'm sure was considered the height of ridiculousness...but since Einstein's work on general relativity we know that the space we live in is governed by exactly that sort of geometry! I could go on and on, but the point is that I would not belittle any area of math as "just recreational" because almost all of it seems to find surprising applications somehow and sometime. (I wrote a story about how weird that is, in fact. See here.)
Thanks for writing...I hope my answer was not too long. Please do feel free to ask for additional explanations if I have not been clear.
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